
"""simple docstring"""






def __magic_name__	(				_lowerCamelCase:		int			)					->						None:

	'''simple docstring'''




	lowerCAmelCase												=  generate_pascal_triangle(_lowerCamelCase			)
	for row_idx in range(_lowerCamelCase			):
		# Print left spaces
		for _ in range(num_rows - row_idx - 1			):
			print(end=''' '''			)
		# Print row values
		for col_idx in range(row_idx + 1			):
			if col_idx != row_idx:
				print(triangle[row_idx][col_idx],					end=''' '''			)
			else:
				print(triangle[row_idx][col_idx],					end=''''''			)
		print()




def __magic_name__	(				_lowerCamelCase:		int			)					->						list[list[int]]:

	'''simple docstring'''




	if not isinstance(_lowerCamelCase,					_lowerCamelCase			):
		raise TypeError('''The input value of \'num_rows\' should be \'int\''''			)

	if num_rows == 0:
		return []
	elif num_rows < 0:
		raise ValueError(
		    '''The input value of \'num_rows\' should be greater than or equal to 0'''			)

	lowerCAmelCase												=  []
	for current_row_idx in range(_lowerCamelCase			):
		lowerCAmelCase												=  populate_current_row(_lowerCamelCase,					_lowerCamelCase			)
		triangle.append(_lowerCamelCase			)
	return triangle




def __magic_name__	(				_lowerCamelCase:		list[list[int]],					_lowerCamelCase:		int			)					->						list[int]:

	'''simple docstring'''




	lowerCAmelCase												=  [-1] * (current_row_idx + 1)
	# first and last elements of current row are equal to 1
	lowerCAmelCase							,       lowerCAmelCase												=  1, 1
	for current_col_idx in range(1,					_lowerCamelCase			):
		calculate_current_element(
		    _lowerCamelCase,					_lowerCamelCase,					_lowerCamelCase,					_lowerCamelCase			)
	return current_row




def __magic_name__	(				_lowerCamelCase:		list[list[int]],					_lowerCamelCase:		list[int],					_lowerCamelCase:		int,					_lowerCamelCase:		int,					)					->						None:

	'''simple docstring'''




	lowerCAmelCase												=  triangle[current_row_idx - 1][current_col_idx - 1]
	lowerCAmelCase												=  triangle[current_row_idx - 1][current_col_idx]
	lowerCAmelCase												=  above_to_left_elt + above_to_right_elt




def __magic_name__	(				_lowerCamelCase:		int			)					->						list[list[int]]:

	'''simple docstring'''




	if not isinstance(_lowerCamelCase,					_lowerCamelCase			):
		raise TypeError('''The input value of \'num_rows\' should be \'int\''''			)

	if num_rows == 0:
		return []
	elif num_rows < 0:
		raise ValueError(
		    '''The input value of \'num_rows\' should be greater than or equal to 0'''			)

	lowerCAmelCase												=  [[1]]

	for row_index in range(1,					_lowerCamelCase			):
		lowerCAmelCase												=  [0] + result[-1] + [0]
		lowerCAmelCase												=  row_index + 1
		# Calculate the number of distinct elements in a row
		lowerCAmelCase												=  sum(divmod(_lowerCamelCase,					2			)			)
		lowerCAmelCase												=  [
		    temp_row[i - 1] + temp_row[i] for i in range(1,					distinct_elements + 1			)
		]
		lowerCAmelCase												=  row_first_half[: (row_index + 1) // 2]
		row_second_half.reverse()
		lowerCAmelCase												=  row_first_half + row_second_half
		result.append(_lowerCamelCase			)

	return result




def __magic_name__	(				)					->						None:

	'''simple docstring'''




	from collections.abc import Callable
	from timeit import timeit

	def benchmark_a_function(_lowerCamelCase:		Callable,					_lowerCamelCase:		int			) -> None:
		lowerCAmelCase												=  F"""{func.__name__}({value})"""
		lowerCAmelCase												=  timeit(F"""__main__.{call}""",					setup='''import __main__'''			)
		# print(f"{call:38} = {func(value)} -- {timing:.4f} seconds")
		print(F"""{call:38} -- {timing:.4f} seconds"""			)

	for value in range(15			):  # (1, 7, 14):
		for func in (generate_pascal_triangle, generate_pascal_triangle_optimized):
			benchmark_a_function(_lowerCamelCase,					_lowerCamelCase			)
		print()


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					benchmark()




"""simple docstring"""






from itertools import count




def __magic_name__	(				_lowerCamelCase:		int = 50			)					->						int:

	'''simple docstring'''




	lowerCAmelCase												=  [1] * min_block_length

	for n in count(_lowerCamelCase			):
		fill_count_functions.append(1			)

		for block_length in range(_lowerCamelCase,					n + 1			):
			for block_start in range(n - block_length			):
				fill_count_functions[n] += fill_count_functions[
				    n - block_start - block_length - 1
				]

			fill_count_functions[n] += 1

		if fill_count_functions[n] > 1_000_000:
			break

	return n


if __name__ == "__main__":
					print(f"""{solution() = }""")


